The secretary problem is a bad way to formalise the problem of dating and the stopping rule doesn't work in that context, for a very simple reason.

The secretary problem is a solution to the problem of having to make repeated choices. In essence it's a solution to the problem of having to choose a secretary every morning for the day. You can even say it's a solution to the problem of a computer process spending a few seconds consuming an infinite stream at the top of every hour for the next hour. It's not a solution to the problem of making a (hopefully) unique choice.